Pattern Synonyms
Most language entities in Haskell can be named so that they can be abbreviated instead of written out in full. This proposal provides the same power for patterns. See the implementation page for implementation details.
See the PatternSynonyms label.
Motivating example
Here is a simple representation of types
data Type = App String [Type]
Using this representations the arrow type looks like App ">" [t1, t2]
.
Here are functions that collect all argument types of nested arrows and recognize the Int
type:
collectArgs :: Type > [Type]
collectArgs (App ">" [t1, t2]) = t1 : collectArgs t2
collectArgs _ = []
isInt (App "Int" []) = True
isInt _ = False
Matching on App
directly is both hard to read and error prone to write.
The proposal is to introduce a way to give patterns names:
pattern Arrow t1 t2 = App ">" [t1, t2]
pattern Int = App "Int" []
And now we can write
collectArgs :: Type > [Type]
collectArgs (Arrow t1 t2) = t1 : collectArgs t2
collectArgs _ = []
isInt Int = True
isInt _ = False
Here is a second example from pigworker on Reddit. Your basic sumsofproducts functors can be built from this kit.
newtype K a x = K a
newtype I x = I x
newtype (:+:) f g x = Sum (Either (f x) (g x))
newtype (:*:) f g x = Prod (f x, g x)
and then you can make recursive datatypes via
newtype Fix f = In (f (Fix f))
e.g.,
type Tree = Fix (K () :+: (I :*: I))
and you can get useful generic operations cheaply because the functors in the kit are all Traversable
, admit a partial zip operation, etc.
You can define friendly constructors for use in expressions
leaf :: Tree
leaf = In (Sum (Left (K ())))
node :: Tree > Tree > Tree
node l r = In (Sum (Right (Prod (I l, I r))))
but any Tree
specific pattern matching code you write will be wide and obscure. Turning these definitions into pattern synonyms means you can have both readable typespecific programs and handy generics without marshalling your data between views.
Unidirectional (patternonly) synonyms
The simplest form of pattern synonyms is the one from the examples above. The grammar rule is:
pattern
conid varid_{1} ... varid_{n} <
pat
pattern
varid_{1} consym varid_{2} <
pat
 Each of the variables on the left hand side must occur exactly once on the right hand side
 Pattern synonyms are not allowed to be recursive. Cf. type synonyms.
There have been several proposals for the syntax of defining patternonly synonyms:


Pattern synonyms can be exported and imported by prefixing the conid with the keyword pattern
:
module Foo (pattern Arrow) where ...
This is required because pattern synonyms are in the namespace of constructors, so it's perfectly valid to have
data P = C
pattern P = 42
You may also give a type signature for a pattern, but as with most other type signatures in Haskell it is optional:
pattern
conid ::
type
E.g.
pattern Arrow :: Type > Type > Type
pattern Arrow t1 t2 < App ">" [t1, t2]
Together with ViewPatterns we can now create patterns that look like regular patterns to match on existing (perhaps abstract) types in new ways:
import qualified Data.Sequence as Seq
pattern Empty < (Seq.viewl > Seq.EmptyL)
pattern x :< xs < (Seq.viewl > x Seq.:< xs)
pattern xs :> x < (Seq.viewr > xs Seq.:> x)
Simplybidirectional pattern synonyms
In cases where pat is in the intersection of the grammars for patterns and expressions (i.e. is valid both as an expression and a pattern), the pattern synonym can be made bidirectional, and can be used in expression contexts as well. Bidirectional pattern synonyms have the following syntax:
pattern
conid varid_{1} ... varid_{n} =
pat
pattern
varid_{1} consym varid_{2} =
pat
For example, the following two pattern synonym definitions are rejected, because they are not bidirectional (but they would be valid as patternonly synonyms)
pattern ThirdElem x = _:_:x:_
pattern Snd y = (x, y)
since the righthand side is not a closed expression of {x} and {y} respectively.
In contrast, the pattern synonyms for Arrow and Int above are bidirectional, so you can e.g. write:
arrows :: [Type] > Type > Type
arrows = flip $ foldr Arrow
Explicitlybidirectional pattern synonyms
What if you want to use Succ
in an expression:
pattern Succ n < n1  let n = n1 1, n >= 0
It's clearly impossible since its expansion is a pattern that has no meaning as an expression. Nevertheless, if we want to make what looks like a constructor for a type we will often want to use it in both patterns and expressions. This is the rationale for the most complicated synonyms, the bidirectional ones. They provide two expansions, one for patterns and one for expressions.
pattern
conid varid_{1} ... varid_{n} <
pat where
cfunlhs rhs
where cfunlhs is like funlhs, except that the functions symbol is a conid instead of a varid.
Example, using ViewPatterns:
pattern Succ n < ((\x > (x1) <$ guard (x > 0)) > Just n)
where
Succ n = n + 1
The first part as is before and describes the expansion of the synonym in patterns. The second part describes the expansion in expressions.
fac (Succ n) = Succ n * fac n
fac 0 = 1
Associated pattern synonyms
Just like data types and type synonyms can be part of a class declaration, it would be possible to have pattern synonyms as well.
Example:
class ListLike l where
pattern Nil :: l a
pattern Cons :: a > l a > l a
isNil :: l a > Bool
isNil Nil = True
isNil (Cons _ _) = False
append :: l a > l a > l a
instance ListLike [] where
pattern Nil = []
pattern Cons x xs = x:xs
append = (++)
headOf :: (ListLike l) => l a > Maybe a
headOf Nil = Nothing
headOf (Cons x _) = Just x
One could go one step further and leave out the pattern
keyword to obtain associated constructors, which are required to be bidirectional. The capitalized identifier would indicate that a pattern synonym is being defined. For complicated cases one could resort to the where
syntax (shown above).
TODO: Syntax for associated pattern synonym declarations to discern between patternonly and bidirectional pattern synonyms
Static semantics
A unidirectional pattern synonym declaration has the form
pattern P var1 var2 ... varN < pat
The formal pattern synonym arguments var1
, var2
, ..., varN
are brought
into scope by the pattern pat on the righthand side. The declaration
brings the name P
as a pattern synonym into the modulelevel scope.
The pattern synonym P
is assigned a pattern type of the form
pattern P :: CReq => CProv => t1 > t2 > ... > tN > t
where t1
, ..., tN
are the types of the parameters var1
, ..., varN
, t
is the simple type (with no context) of the thing getting matched, and CReq
and CProv
are type contexts.
CProv
can be omitted if it is empty. If CReq
is empty, but CProv
is not, ()
is used. The following example shows cases:
data Showable where
MkShowable :: (Show a) => a > Showable
 Required context is empty, but provided context is not
pattern Sh :: () => (Show a) => a > Showable
pattern Sh x < MkShowable x
 Provided context is empty
pattern One :: (Num a, Eq a) => a
pattern One < 1
A pattern synonym can be used in a pattern if the
instatiated (monomorphic) type satisfies the constraints of
CReq
. In this case, it extends the context available in the
righthand side of the match with CProv
, just like how an
existentiallytyped data constructor can extend the context.
As with function and variable types, the pattern type signature can be inferred, or it can be explicitly written out on the program.
Here's a more complex example. Let's look at the following definition:
{# LANGUAGE PatternSynonyms, GADTs, ViewPatterns #}
module ShouldCompile where
data T a where
MkT :: (Eq b) => a > b > T a
f :: (Show a) => a > Bool
pattern P x < MkT (f > True) x
Here, the inferred type of P
is
pattern P :: (Show a) => (Eq b) => b > T a
A bidirectional pattern synonym declaration has the form
pattern P var1 var2 ... varN = pat
where both of the following are welltyped declarations:
pattern P1 var1 var2 ... varN < pat
P2 = \var1 var2 ... varN > pat
In this case, the pattern type of P
is simply the pattern type
of P1
, and its expression type is the type of P2
. The name P
is brought into the modulelevel scope both as a pattern synonym and
as an expression.
Dynamic semantics
A pattern synonym occurance in a pattern is evaluated by first matching against the pattern synonym itself, and then on the argument patterns. For example, given the following definitions:
pattern P x y < [x, y]
f (P True True) = True
f _ = False
g [True, True] = True
g _ = False
the behaviour of f
is the same as
f [x, y]  True < x, True < y = True
f _ = False
Because of this, the eagerness of f
and g
differ:
*Main> f (False:undefined)
*** Exception: Prelude.undefined
*Main> g (False:undefined)
False
This is because we generate the matching function at the definition site.
Typed pattern synonyms
So far patterns only had syntactic meaning. In comparison Ωmega has typed pattern synonyms, so they become first class values. For bidirectional pattern synonyms this seems to be the case
data Nat = Z  S Nat deriving Show
pattern Ess p = S p
And it works:
*Main> map S [Z, Z, S Z]
[S Z,S Z,S (S Z)]
*Main> map Ess [Z, Z, S Z]
[S Z,S Z,S (S Z)]
Branching patternonly synonyms
*N.B. this is a speculative suggestion! *
Sometimes you want to match against several summands of an ADT simultaneously. E.g. in a data type of potentially unbounded natural numbers:
data Nat = Zero  Succ Nat
type UNat = Maybe Nat  Nothing meaning unbounded
Conceptually Nothing
means infinite, so it makes sense to interpret it as a successor of something. We wish it to have a predecessor just like Just (Succ Zero)
!
I suggest branching pattern synonyms for this purpose:
pattern S pred < pred@Nothing  pred@(Just a < Just (Succ a))
pattern Z = Just Zero
Here pred@(Just a < Just (Succ a))
means that the pattern invocation S pred
matches against Just (Succ a)
and  if successful  binds Just a
to pred
.
This means we can syntactically address unbound naturals just like bounded ones:
greetTimes :: UNat > String > IO ()
greetTimes Z _ = return ()
greetTimes (S rest) message = putStrLn message >> greetTimes rest message
As a nice collateral win this proposal handles pattern Name name < Person name workplace  Dog name vet
too.
Record Pattern Synonyms
See PatternSynonyms/RecordPatternSynonyms
Associating synonyms with types
See PatternSynonyms/AssociatingSynonyms